New leading contributions to non-gaussianity in single field inflation

We compute the bispectrum of primordial density perturbations in CMB to second order in the slow-roll parameters of single field inflation. We correct previous results and found that next-to-leading order corrections can be of the same order as the leading order result in a large class of models, including hilltop inflation.

💡 Research Summary

The paper presents a comprehensive calculation of the primordial bispectrum generated during single‑field slow‑roll inflation, extending the analysis to second order in the Hubble‑flow (slow‑roll) parameters. Starting from the standard Einstein‑Hilbert action with a scalar inflaton, the authors adopt the ADM formalism and work in the ζ‑gauge (where the inflaton fluctuation is set to zero) to derive the quadratic and cubic actions for the curvature perturbation ζ. The cubic action contains a rich set of interaction operators proportional to combinations of ε₁, ε₂, ε₃, etc., including terms proportional to the equations of motion (EOM) which, although often discarded, contribute via delta‑function insertions in the Schwinger‑Keldysh path‑integral formalism.

Using the in‑in (Schwinger‑Keldysh) formalism, the three‑point function ⟨ζζζ⟩ is evaluated at tree level. The contour consists of forward (C⁺) and backward (C⁻) time branches, leading to two copies of the field (ζ⁺, ζ⁻) and four Green functions G_{ab}. Wick contractions are performed systematically, and the time integrals are carried out with careful treatment of the iε prescription that selects the Bunch‑Davies vacuum. A crucial observation is that infrared logarithmic growth of the propagator in de Sitter space yields terms proportional to ln(k_t/k_*) where k_t = k₁ + k₂ + k₃. Because the logarithm can be as large as the number of e‑folds (≈60), these NLO contributions can compensate the usual ε suppression.

The bispectrum amplitude is expressed as a sum of nine distinct contributions, one of which originates from the EOM‑proportional operator and reproduces the gauge‑shift term known from the Hamiltonian approach. After summing all pieces, the authors obtain a momentum‑dependent f_NL(k₁,k₂,k₃) that reduces to the familiar local shape (constant) at leading order but acquires a sizable logarithmic dependence at next‑to‑leading order. In the squeezed limit (k₁ ≪ k₂ ≈ k₃) the logarithmic term dominates, leading to an enhancement of f_NL relative to the LO estimate of ≈2×10⁻².

To illustrate the phenomenological impact, the authors evaluate the result for a hill‑top potential V(φ)=V₀