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The Physical Squeezed Limit: Consistency Relations at Order q^2

Paolo Creminelli, Ashley Perko, Leonardo Senatore, Marko Simonović, Gabriele Trevisan

TL;DR

This work shows that in single-field inflation the squeezed limit at order $q^2$ corresponds to the long mode inducing spatial curvature, allowing a curved-FRW description for the short modes. By mapping the long mode to curvature and computing the short-mode two-point function in this curved background, the authors derive and test a curvature-derivative consistency relation for the three-point function, $\langle \zeta_{\vec{q}} \zeta_{\vec{k}_1} \zeta_{\vec{k}_2} \rangle'_{q\to0,avg} = P_{\zeta}(q) \frac{2}{3} q^2 \frac{\partial}{\partial\kappa} \langle \zeta_{\vec{k}_1} \zeta_{-\vec{k}_1} \rangle'_{\kappa}$. They perform explicit EFTI checks with reduced $c_s$ (and related models such as Ghost Inflation and Khronon Inflation), showing consistency at both leading and all orders in $c_s$, and clarifying how non-Gaussianity scales as $f_{NL} \sim (k_{\rm ph,f}/H)^2$. The results provide a conceptually transparent link between large-scale curvature effects and small-scale statistics, while highlighting the limitations to $\mathcal{O}(q^2)$ where a classical long mode description remains valid.

Abstract

In single-field models of inflation the effect of a long mode with momentum q reduces to a diffeomorphism at zeroth and first order in q. This gives the well-known consistency relations for the n-point functions. At order q^2 the long mode has a physical effect on the short ones, since it induces curvature, and we expect that this effect is the same as being in a curved FRW universe. In this paper we verify this intuition in various examples of the three-point function, whose behaviour at order q^2 can be written in terms of the power spectrum in a curved universe. This gives a simple alternative understanding of the level of non-Gaussianity in single-field models. Non-Gaussianity is always parametrically enhanced when modes freeze at a physical scale k_{ph, f} shorter than H: f_{NL} \sim (k_{ph, f}/H)^2.

The Physical Squeezed Limit: Consistency Relations at Order q^2

TL;DR

This work shows that in single-field inflation the squeezed limit at order corresponds to the long mode inducing spatial curvature, allowing a curved-FRW description for the short modes. By mapping the long mode to curvature and computing the short-mode two-point function in this curved background, the authors derive and test a curvature-derivative consistency relation for the three-point function, . They perform explicit EFTI checks with reduced (and related models such as Ghost Inflation and Khronon Inflation), showing consistency at both leading and all orders in , and clarifying how non-Gaussianity scales as . The results provide a conceptually transparent link between large-scale curvature effects and small-scale statistics, while highlighting the limitations to where a classical long mode description remains valid.

Abstract

In single-field models of inflation the effect of a long mode with momentum q reduces to a diffeomorphism at zeroth and first order in q. This gives the well-known consistency relations for the n-point functions. At order q^2 the long mode has a physical effect on the short ones, since it induces curvature, and we expect that this effect is the same as being in a curved FRW universe. In this paper we verify this intuition in various examples of the three-point function, whose behaviour at order q^2 can be written in terms of the power spectrum in a curved universe. This gives a simple alternative understanding of the level of non-Gaussianity in single-field models. Non-Gaussianity is always parametrically enhanced when modes freeze at a physical scale k_{ph, f} shorter than H: f_{NL} \sim (k_{ph, f}/H)^2.

Paper Structure

This paper contains 11 sections, 101 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. From $\zeta$-gauge to a curved FRW universe
  3. Checks of the consistency relation
  4. Quadratic action for short modes in a curved FRW universe
  5. The two-point function at leading order in $1/c_s^2$
  6. The two-point function at all orders in $c_s$
  7. Conclusions
  8. Full diffeomorphism from flat to curved EFTI
  9. K-inflation
  10. Ghost Inflation
  11. Khronon Inflation

Figures (1)

  • Figure 1: Parametric dependence of non-Gaussianity with the freezing scale. On the left the freezing scale is the Hubble scale. On the right the freezing scale is parametrically shorter than Hubble: in this case the effect of the curvature induced by the long mode is bigger, since it has to be compared with $H^2$ in the Friedmann equation.