  In single field inflationary models, preheating refers to the phase that
immediately follows inflation, but precedes the epoch of reheating. During this
phase, the inflaton typically oscillates at the bottom of its potential and
gradually transfers its energy to radiation. At the same time, the amplitude of
the fields coupled to the inflaton may undergo parametric resonance and, as a
consequence, explosive particle production can take place. A priori, these
phenomena could lead to an amplification of the super-Hubble scale curvature
perturbations which, in turn, would modify the standard inflationary
predictions. However, remarkably, it has been shown that, although the
Mukhanov-Sasaki variable does undergo narrow parametric instability during
preheating, the amplitude of the corresponding super-Hubble curvature
perturbations remain constant. Therefore, in single field models, metric
preheating does not affect the power spectrum of the large scale perturbations.
In this article, we investigate the corresponding effect on the scalar
bi-spectrum. Using the Maldacena's formalism, we analytically show that, for
modes of cosmological interest, the contributions to the scalar bi-spectrum as
the curvature perturbations evolve on super-Hubble scales during preheating is
completely negligible. Specifically, we illustrate that, certain terms in the
third order action governing the curvature perturbations which may naively be
expected to contribute significantly are exactly canceled by other
contributions to the bi-spectrum. We corroborate selected analytical results by
numerical investigations. We conclude with a brief discussion of the results we
have obtained.
