Inflationary soft theorems revisited: a generalized consistency relation

Abstract

We reconsider the derivation of soft theorems associated with\nnonlinearly-realized symmetries in cosmology. Utilizing the path integral, we\nderive a generalized consistency relation that relates a squeezed $(N+1)$-point\ncorrelation function to an $N$-point function, where the relevant soft mode is\nat early rather than late time. This generalized (early-late-time) version has\nwider applicability than the standard consistency relation where all modes are\nevaluated at late times. We elucidate the conditions under which the latter\nfollows from the former. A key ingredient is the physical mode condition: that\nthe nonlinear part of the symmetry transformation must match the time\ndependence of the dominant, long wavelength physical mode. This is closely\nrelated to, but distinct from, the adiabatic mode condition. Our derivation\nsheds light on a number of otherwise puzzling features of the standard\nconsistency relation: (1) the underlying nonlinearly-realized symmetries (such\nas dilation and special conformal transformation SCT) originate as residual\ngauge redundancies, yet the consistency relation has physical content---for\ninstance, it can be violated; (2) the standard consistency relation is known to\nfail in ultra-slow-roll inflation, but since dilation and SCT remain good\nsymmetries, there should be a replacement for the standard relation; (3) in\nlarge scale structure applications, it is known that the standard consistency\nrelation breaks down if the long wavelength power spectrum is too blue. The\nearly-late-time consistency relation helps address these puzzles. We introduce\na toy model where explicit checks of this generalized consistency relation are\nsimple to carry out. Our methodology can be adapted to cases where violations\nof the standard consistency relation involve additional light degrees of\nfreedom beyond the inflaton.\n