Conformal consistency relations for single-field inflation

Abstract

We generalize the single-field consistency relations to capture not only the leading term in the squeezed limit-going as 1/q 3 , where q is the small wavevector-but also the subleading one, going as 1/q 2 . This term, for an (n + 1)-point function, is fixed in terms of the variation of the n-point function under a special conformal transformation; this parallels the fact that the 1/q 3 term is related with the scale dependence of the n-point function. For the squeezed limit of the 3-point function, this conformal consistency relation implies that there are no terms going as 1/q 2 . We verify that the squeezed limit of the 4-point function is related to the conformal variation of the 3-point function both in the case of canonical slow-roll inflation and in models with reduced speed of sound. In the second case the conformal consistency conditions capture, at the level of observables, the relation among operators induced by the non-linear realization of Lorentz invariance in the Lagrangian. These results mean that, in any single-field model, primordial correlation functions of ζ are endowed with an SO(4, 1) symmetry, with dilations and special conformal transformations nonlinearly realized by ζ. We also verify the conformal consistency relations for any n-point function in models with a modulation of the inflaton potential, where the scale dependence is not negligible. Finally, we generalize (some of) the consistency relations involving tensors and soft internal momenta.

On separate universes

“…where Φ F , Ψ F are simply related to the scalar potentials in the conformal Newtonian gauge through 5) and an analogous relation holds for Ψ F in terms of Ψ. Note that Φ F , Ψ F involve the tracefree part of the second spatial derivatives of Φ and Ψ, i.e.…”

An effective theory for holographic RG flows

“…It would be very interesting to understand the universality of anomaloy matching in complete generality, without the slow-flow assumption. It might also be interesting to study the constraints from AdS/CFT Ward identities [37][38][39][40] related to the consistency relation [20,21,[41][42][43] in cosmology. The effective theory we develop may have broader applications for the study of holographic phenomenology [27,28], where our methods could be used to derive both the universal anomaly matching terms and the conformally invariant terms for light dilatons.…”

How Gaussian can our Universe be?

“…C, more precisely the fact that 21) and averaging over the long-wavelength perturbations, we reproduce the η part of Maldacena's bispectrum in the squeezed limit, up to and including O(k 2 /k 2 s ). Summing this to the other contribution (noting that the first term in Eq.…”