Inflationary soft theorems revisited: A generalized consistency relation

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.

arXiv comment74 pages (main text: 38 pages, appendix: 30 pages), 3 figures, 2 tables

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