The squeezed limit of the solid inflation three-point function

TL;DR

Solid inflation predicts a highly anisotropic squeezed-limit three-point function for scalars, with a pure quadrupolar angular dependence and potentially large amplitude tied to $F_Y/F$ and $\epsilon$. The authors present a simple background-field derivation that treats a long-wavelength mode as a background that renormalizes the short-mode dynamics, thereby reproducing the squeezed limit without relying on the standard consistency relations; they show the long-wavelength background shifts the longitudinal phonon speed as $\tilde{c}_L^2 = c_L^2 + \frac{8}{9}\frac{F_Y}{F}\frac{1}{\epsilon}(1-3 \cos^2\theta)\zeta_\ell$, and compute the corresponding $\langle \zeta \zeta \zeta \rangle$ in the squeezed limit: $\langle \zeta_{\vec{q} \to 0} \zeta_{\vec{k}} \zeta_{-\vec{k}} \rangle' \simeq -\frac{20}{9}\frac{F_Y}{F}\frac{1}{\epsilon c_L^2}(1-3\cos^2\theta) P_\zeta(q)P_\zeta(k)$. Extending the logic to vector and tensor modes yields additional squeezed-limit relations with specific angular and polarization structures, all consistent with a general angular-sum rule. The work provides a practical, general calculational framework for non-standard inflationary models with similar symmetry-breaking patterns and discusses implications for higher-point functions and quantum vs classical long modes.

Abstract

The recently proposed model of 'solid inflation' features a peculiar three-point function for scalar perturbations with an anisotropic, purely quadrupolar, squeezed limit. We confirm this result as well as the overall amplitude of the three point-function via an extremely simple computation, where we focus on the squeezed limit from the start and follow the standard logic adopted in deriving the consistency relations. Our system violates the consistency relations, but in the squeezed limit the three-point function can still be traded for a background-dependent two-point function, which is immediate to compute. Additionally, we use these simple methods to derive some new results - namely, certain squeezed limits of the three-point correlators involving vector and tensor perturbations as well.