Single-Field Consistency relation and $\delta N$-Formalism

Abstract

According to the equivalence principal, the long wavelength perturbations must not have any dynamical effect on the short scale physics up to ${\cal O} (k_L^2/k_s^2)$. Their effect can be always absorbed to a coordinate transformation locally. So any physical effect of such a perturbation appears only on scales larger than the scale of the perturbation. The bispectrum in the squeezed limit of the curvature perturbation in single-field slow-roll inflation is a good example, where the long wavelength effect is encoded in the spectral index through Maldacena's consistency relation. This implies that one should be able to derive the bispectrum in the squeezed limit without resorting to the in-in formalism in which one computes perturbative corrections field-theoretically. In this short paper, we show that the $\delta N$ formalism as it is, or more generically the separate universe approach, when applied carefully can indeed lead to the correct result for the bispectrum in the squeezed limit. Hence despite the common belief that the $\delta N$ formalism is incapable of recovering the consistency relation within itself, it is in fact self-contained and consistent.