Conformal consistency relations for single-field inflation (1203.4595v2)

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 \zeta are endowed with an SO(4,1) symmetry, with dilations and special conformal transformations non-linearly realized by \zeta. 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.

• The paper demonstrates that subleading 1/q² contributions in the squeezed limit arise from the variation of n-point functions under special conformal transformations.
• It shows that while these corrections are absent in the 3-point function, they emerge prominently in the 4-point function across various single-field models.
• The study links SO(4,1) symmetry to both dilation and special conformal invariance, offering refined predictions for observable primordial non-Gaussianities.

Conformal Consistency Relations for Single-Field Inflation

The paper "Conformal consistency relations for single-field inflation" by Paolo Creminelli, Jorge Norena, and Marko Simonović presents a detailed exploration of the conformal properties of primordial perturbations within single-field inflation models. The authors aim to elucidate the role of conformal transformations, particularly emphasizing the inclusion of subleading terms in the squeezed limit of correlation functions, which extend beyond the traditional leading-order terms.

The primary contribution of this paper is the generalization of single-field consistency relations to account for subleading contributions—specifically, those proportional to $1/q^2$ in the squeezed limit, where $q$ is the small wavevector. Traditionally, the $1/q^3$ terms have been associated with dilation invariance, which relates to the scale dependence of correlation functions. The authors demonstrate that the new $1/q^2$ term is intricately linked to the variation of the $n$-point function under a special conformal transformation.

For the 3-point function in the squeezed limit, it is shown that the $1/q^2$ contribution is absent, while for the 4-point function, the subleading term emerges from the conformal variation of the lower-order 3-point function. This correlation holds in both canonical slow-roll scenarios and models characterized by a reduced speed of sound, indicating the robustness of these analytics across various single-field configurations.

A significant theoretical implication of these findings is the symmetry that emerges in any single-field inflationary scenario: the primordial correlation functions are governed by an $SO(4,1)$ symmetry. This symmetry arises from the non-linear realization of both dilations and special conformal transformations. The paper offers a mathematical formalism that computes the effect of background perturbations as transformations in coordinate space, reflecting on how these transformations affect observable correlation functions.

Furthermore, the paper extends these consistency relations to accommodate scenarios with non-negligible scale dependence, exemplified by models with resonant non-Gaussianity. This extension integrates both leading and subleading terms, encapsulating the complete range of potential behavior in the squeezed limit.

In discussing tensor modes, the authors derive consistency relations involving one long-wavelength graviton mode combined with scalar perturbations, while noting the limitations in extending these formulations to scenarios with multiple short tensor modes due to the inherent non-linear nature of these interactions.

The paper closes by contemplating on the impact of these conformal consistency relations, not only on theoretical grounds but also in practical applications such as observational cosmology, where they could offer insights into the primordial universe by dissecting the intricate symmetries encoded in the correlation functions of cosmic inflation.

In conclusion, this work not only broadens our understanding of conformal symmetries in the context of single-field inflation but also sets the stage for refining theoretical predictions related to primordial non-Gaussianities that might be probed with future cosmic surveys, potentially offering new pathways to elucidate the fundamental mechanics of early-universe physics.