Inflationary soft theorems revisited: A generalized consistency relation

TL;DR

This work derives a generalized early-late-time Ward identity for soft theorems in cosmology, linking an initial-time soft insertion to late-time correlators via a path-integral approach. It introduces the physical mode condition to determine when unequal-time results reduce to standard late-time relations, and distinguishes it from the adiabatic mode condition. Applied to inflation, the framework clarifies why standard Maldacena-type relations can fail in ultra-slow-roll and how slow-roll attractors recover the familiar equal-time identities. A versatile toy model demonstrates the broad applicability and helps resolve puzzles about symmetry origins, initial states, and potential violations. Overall, the paper provides a robust, general mechanism to understand and test cosmological soft theorems beyond the traditional late-time paradigm.

Abstract

We reconsider the derivation of soft theorems associated with nonlinearly-realized symmetries in cosmology. Utilizing the path integral, we derive a generalized consistency relation that relates a squeezed $(N+1)$-point correlation function to an $N$-point function, where the relevant soft mode is at early rather than late time. This generalized (early-late-time) version has wider applicability than the standard consistency relation where all modes are evaluated at late times. We elucidate the conditions under which the latter follows from the former. A key ingredient is the physical mode condition: that the nonlinear part of the symmetry transformation must match the time dependence of the dominant, long wavelength physical mode. This is closely related to, but distinct from, the adiabatic mode condition. Our derivation sheds light on a number of otherwise puzzling features of the standard consistency relation: (1) the underlying nonlinearly-realized symmetries (such as dilation and special conformal transformation SCT) originate as residual gauge redundancies, yet the consistency relation has physical content---for instance, it can be violated; (2) the standard consistency relation is known to fail in ultra-slow-roll inflation, but since dilation and SCT remain good symmetries, there should be a replacement for the standard relation; (3) in large scale structure applications, it is known that the standard consistency relation breaks down if the long wavelength power spectrum is too blue. The early-late-time consistency relation helps address these puzzles. We introduce a toy model where explicit checks of this generalized consistency relation are simple to carry out. Our methodology can be adapted to cases where violations of the standard consistency relation involve additional light degrees of freedom beyond the inflaton.

Figures (3)

• Figure 1: The new identity relates $(N+1)$ point functions, with the soft mode inserted in the far past, to $N$ point functions.
• Figure 2: The usual soft theorems relates a squeezed $(N+1)$-point equal time correlator to an $N$-point correlator.
• Figure 3: Perturbative proof of the use of physical mode. $L_{(3)}[ \delta_{\rm NL}u_q, u_{k_1}, u_{k_2} ]$ is the interaction vertex inside the in-in time integral $\bigl\langle\zeta_{\vec{q}}(\tau) \zeta_{\vec{k}_1}(\tau)\zeta_{\vec{k}_2}(\tau)\bigr\rangle' = 2 {\rm Im} ( u^*_q(\tau) u^*_{k_1}(\tau) u^*_{k_2}(\tau) \int^{\tau} {\rm d} \tau' L^{(3)}[ u_q, u_{k_1}, u_{k_2} ](\tau') ) +{\rm perms.}$