A universal bound on N-point correlations from inflation

TL;DR

This work proves a universal bound on inflationary N-point correlations, showing $\tau_{NL} \ge \left(\frac{6}{5} f_{NL}\right)^2$ for any non-Gaussian initial conditions. The authors present a general positivity-based argument using a long-wavelength modulation of small-scale power and a positive-definite covariance between $\zeta$ and the estimated power, valid in the squeezed limit; they also discuss finite-volume corrections and the meaning of apparent violations when $f_{NL}$ and $\tau_{NL}$ are defined at fixed scales. They analyze estimator behavior for ideal and Planck-like experiments, noting that observed violations can arise from estimator effects or translation-invariance assumptions rather than true Hubble-volume violations. As a concrete check, they study the ungaussiton model, showing the SY bound holds exactly and clarifying the limitations of naive $\tau_{NL}$-$f_{NL}$ scaling at large $f_{NL}$. Overall, the SY inequality is presented as a universal positivity constraint with important implications for interpreting non-Gaussian signals in the CMB and large-scale structure.

Abstract

Models of inflation in which non-Gaussianity is generated outside the horizon, such as curvaton models, generate distinctive higher-order correlation functions in the CMB and other cosmological observables. Testing for violation of the Suyama-Yamaguchi inequality tauNL >= (6/5 fNL)^2, where fNL and tauNL denote the amplitude of the three-point and four-point functions in certain limits, has been proposed as a way to distinguish qualitative classes of models. This inequality has been proved for a wide range of models, but only weaker versions have been proved in general. In this paper, we give a proof that the Suyama-Yamaguchi inequality is always satisfied. We discuss scenarios in which the inequality may appear to be violated in an experiment such as Planck, and how this apparent violation should be interpreted. We analyze a specific example, the "ungaussiton" model, in which leading-order scaling relations suggest that the Suyama-Yamaguchi inequality is eventually violated, and show that the inequality always holds.