Non-gaussianity of inflationary field perturbations from the field equation

TL;DR

This work derives the tree-level bispectrum of inflaton perturbations directly from the field equations in uniform curvature gauge, demonstrating agreement with Lagrangian-based results. By decomposing δφ into a linear Gaussian part and a quadratic nonlinear part, and employing a retarded Green’s function, the authors compute the three-point function from V'''-type, zero-derivative, and two-derivative contributions. They then connect δφ to the curvature perturbation ζ via the δN formalism to obtain f_NL, showing that in single-field inflation the explicit N_* dependence cancels, leaving slow-roll–driven contributions. The approach provides a robust, Lagrangian-free route to non-Gaussianity, emphasizing that horizon-exit initial conditions plus classical superhorizon evolution yield reliable predictions, with caveats for multi-field or non-standard models.

Abstract

We calculate the tree-level bispectrum of the inflaton field perturbation directly from the field equations, and construct the corresponding f_NL parameter. Our results agree with previous ones derived from the Lagrangian. We argue that quantum theory should only be used to calculate the correlators when they first become classical a few Hubble times after horizon exit, the classical evolution taking over thereafter.