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

TL;DR

This work shows that in a two-field, small-field slow-roll inflation model with canonical kinetic terms, the conventional expectation of negligible primordial non-Gaussianity can be overturned when ζ-series convergence and loop corrections are properly accounted. By computing tree-level and one-loop contributions to the spectrum $P_\zeta$ and bispectrum $B_\zeta$ within a quadratic potential, the authors identify parameter-space regions where $B_\zeta$-loop corrections dominate and yield observable $f_{NL}$, particularly when $P_\zeta$ remains tree-dominated. They classify regimes by the location of the inflationary field value $\phi_*$ (low, intermediate, high) and impose constraints from the COBE normalization, spectral tilt, and required amount of inflation, showing that sizeable $f_{NL}$ is achievable in the intermediate region while remaining consistent with current data. The analysis underscores the necessity of validating the perturbative expansion and ζ-series convergence in δN calculations, and it sets the stage for a companion study of $\tau_{NL}$ in the trispectrum. Overall, the results imply that observable primordial non-Gaussianity may arise within slow-roll, canonical-kinetic-term models when loop effects are properly incorporated, challenging prevailing assumptions in the literature.

Abstract

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_ζof the primordial curvature perturbation ζ, 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_ζand the bispectrum B_ζ. Sizeable values for f_{NL} arise even if ζis 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 ζseries expansion, and its truncation, are valid. 2. we must apply the correct condition for the (possible) loop dominance in B_ζand/or P_ζ. 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.

Figures (6)

• Figure 1: Our small-field slow-roll potential of Eq. (\ref{['pot']}) with $\eta_\phi,\eta_\sigma < 0$. The inflaton starts near the maximum and moves away from the origin following the $\sigma = 0$ trajectory depicted with the solid black line. (This figure has been taken from Ref. alabidi1).
• Figure 2: Contours of $f_{NL}$ in the $r$ vs $|\eta_\sigma|$ plot. The intermediate (high) $\phi_\star$ region is below (above) the LIf line. The WMAP (and also PLANCK) observationally allowed $2\sigma$ range of values for negative $f_{NL}$, $-9 < f_{NL}$, is completely inside the intermediate $\phi_\star$ region. Notice that the LIf line matches almost exactly the $f_{NL} = -1.667$ line.
• Figure A1: Tree-level Feynman-like diagram for $P_\zeta$. The internal dashed line corresponds to a two-point correlator of field perturbations.
• Figure A2: One-loop Feynman-like diagrams for $P_\zeta$. (a). The two internal dashed lines correspond to two-point correlators of field perturbations. (b). The internal dashed lines correspond to a three-point correlator of field perturbations.
• Figure A3: Tree-level Feynman-like diagrams for $B_\zeta$. (a). The two internal dashed lines correspond to two-point correlators of field perturbations. (b). The internal dashed lines correspond to a three-point correlator of field perturbations.