On the Issue of the zeta Series Convergence and Loop Corrections in the Generation of Observable Primordial Non-Gaussianity in Slow-Roll Inflation. Part I: the Bispectrum

We show in this paper that it is possible to attain very high, {\it including observable}, values for the level of non-gaussianity f_{NL} associated with the bispectrum B_\zeta of the primordial curvature perturbation \zeta, in a subclass of small-field {\it slow-roll} models of inflation with canonical kinetic terms. Such a result is obtained by taking care of loop corrections both in the spectrum P_\zeta and the bispectrum B_\zeta. Sizeable values for f_{NL} arise even if \zeta is generated during inflation. Five issues are considered when constraining the available parameter space: 1. we must ensure that we are in a perturbative regime so that the \zeta series expansion, and its truncation, are valid. 2. we must apply the correct condition for the (possible) loop dominance in B_\zeta and/or P_\zeta. 3. we must satisfy the spectrum normalisation condition. 4. we must satisfy the spectral tilt constraint. 5. we must have enough inflation to solve the horizon problem.