A generalized non-Gaussian consistency relation for single field inflation

TL;DR

The paper generalizes Maldacena's squeezed-limit non-Gaussianity to scenarios where the curvature perturbation $ abla$ evolves on super-horizon scales by exploiting a spacetime diffeomorphism plus a curvature reparametrization. In the $oldsymbol{ ext{ε} o 0}$ limit, the squeezed bispectrum can be expressed as $B_ abla(k_1,k_2,k_3) = - P_ abla(k_L) rac{d}{d au} P_ abla(k_S)$, linking the long-mode growth to time evolution of the power spectrum; for ultra slow-roll inflation, this reduces to $B_ abla = 6 P_ abla(k_L) P_ abla(k_S)$, an exact result. Away from ultra slow-roll, with $oldsymbol{ ext{ε}} eq 0$, the symmetry is not exact and the generalized relation becomes approximate, with special USR-like conditions recovering exactness; the paper also discusses extensions to non-canonical models (e.g., varying sound speed $c_s$) and observational implications via conformal Fermi coordinates. Overall, the work provides a unifying symmetry-based framework for squeezed non-Gaussianity beyond standard attractor single-field inflation and clarifies when the standard vanishing of $f_{ m NL}^{ m obs}$ can fail or persist.

Abstract

We show that a perturbed inflationary spacetime, driven by a canonical single scalar field, is invariant under a special class of coordinate transformations together with a field reparametrization of the curvature perturbation in co-moving gauge. This transformation may be used to derive the squeezed limit of the 3-point correlation function of the co-moving curvature perturbations valid in the case that these do not freeze after horizon crossing. This leads to a generalized version of Maldacena's non-Gaussian consistency relation in the sense that the bispectrum squeezed limit is completely determined by spacetime diffeomorphisms. Just as in the case of the standard consistency relation, this result may be understood as the consequence of how long-wavelength modes modulate those of shorter wavelengths. This relation allows one to derive the well known violation to the consistency relation encountered in ultra slow-roll, where curvature perturbations grow exponentially after horizon crossing.