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 ultraslow-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. * 8 We often switch between the singular and plural form for "consistency relation". The singular form refers to the one that holds for all relevant symmetries (such as (2.22)), whereas the plural form counts one for each symmetry separately (such as dilation consistency relation, SCT consistency relation and so on). 9 We follow convention and sometimes refer to the consistency relation as an "identity", but it should be emphasized the consistency relation has non-trivial physical content, as explained above and further in Section 2.2.

The galaxy bias at second order in general relativity with non-Gaussian initial conditions

“…We used the local coordinates introduced in [26], which are equivalent to the conformal Fermi coordinates developed in [29,58]. These local coordinates are based on the non-linearly realised symmetries associated with the consistency relation [39,59,60].…”

Nonperturbative test of consistency relations and their violation

“…Here ≡ ∂/∂η is the derivative with respect to the conformal time η, and H ≡ a /a is the comoving Hubble parameter, with a being the scale factor. The above is a symmetry of the large scale structure dynamics for n having any time dependence, but the adiabatic mode condition [18] dictates that n must match the time-dependence of the linear growth factor, and likewise c should match the corresponding time-dependence of the gravitational potential (see discussions in [10,17] and point 2 below). The consistency relations corresponding to a shift of the gravitational potential by a constant and by a linear gradient are respectively:…”

Spatial curvature at the sound horizon

“…The most famous soft theorem has been derived by Maldacena in [6] and fixes the squeezed bispectrum in terms of the power spectrum and it applies to all attractor, singlefield models of inflation [7]. This first result has been extended to higher n-point functions for primordial scalar, tensor and vector perturbations [8,9,10,11,12,13,14,15,16,17,18,19]. Soft theorems are conveniently interpreted as the consequence of residual, non-linearly realized symmetries associated with adiabatic modes [20,9,21,18,22], namely physical perturbations that are indistinguishable from a change of coordinates in the neighborhood of a point in spacetime.…”

“…To answer the second question, we map the constraints (confidence regions) for c s and c 3 obtained from Planck bispectrum data to the constraints (confidence regions) for B K1 and B K2 . 19 Figure 7 shows the approximate 68%, 95% and 99.7% confidence regions for (B K1 , B K2 ), obtained by mapping the confidence region from the (c s , c 3 ) parameter space. We see that the magnitude of the coefficients is bounded, and an order-of-magnitude estimate is |B K1,2 | 10 6 ∼ 10 7 .…”

“…(A.18)For K > 0 the spectrum of scalar harmonics is ([56]) plm = −p(p + 2)K Y plm , p = 0, 1, .. , and l = 0, .., p . (A 19). and they are given byY plm = (p + 1)Γ(p + l + 2) Γ(p − l + 1) p Kr f (|K|r 2 ) P −l−1/2 p−1/2 1 − K 2 r 2 f 2 (Kr 2 ) Y lm (θ, φ) ,with the same property as (A.18) except that δ pp replaces δ(p − p ).…”