On the squeezed limit of the bispectrum in general single field inflation

TL;DR

This work derives a general, non-approximate integral formula for the squeezed limit of the bispectrum in any Lorentz-invariant single-field inflation model, expressed entirely in terms of the primitive mode functions of the curvature perturbation. It extends the Ganc–Komatsu result beyond canonical kinetic terms and uses it to verify Maldacena’s consistency relation in an exactly solvable class with non-trivial speed of sound, as well as at first and second order in the slow-variation expansion for general single-field and canonical inflation, respectively. The findings demonstrate the robustness of the squeezed-limit relation and provide a practical framework to test single-field dynamics via non-Gaussianity, while highlighting that multi-field scenarios can evade the relation. Overall, the paper clarifies how the squeezed bispectrum encodes the scalar spectral tilt and confirms the consistency relation across a broad range of single-field models.

Abstract

We investigate the consistency relation relating the squeezed limit of the bispectrum to the scalar spectral index in single field models of inflation. We give a simple integral formula for the bispectrum in the squeezed limit in terms of the free mode mode functions of the primordial curvature perturbation, in any Lorentz invariant single field model of inflation and without resorting to any approximation, generalizing a recent result obtained by Ganc and Komatsu in the case of canonical kinetic terms. We use our result to verify the consistency relation in an exactly solvable class of models with a non-trivial speed of sound. We then verify the consistency relation at the first non-trivial order in the slow-varying approximation in general single field inflation (a known result) and at second order in this approximation in canonical single field inflation.