Resonant Trispectrum and a Dozen More Primordial N-point functions

TL;DR

This work develops a gravity-decoupled framework to compute tree-level N-point primordial curvature correlators in resonant inflation up to N roughly 10–20, leveraging a large non-Gaussianity parameter regime. By focusing on the inflaton sector and using in-in perturbation theory, the authors derive a general leading expression for N-point functions with an oscillatory resonant shape, plus first subleading corrections, and verify squeezed-limit consistency relations. They further analyze the growth of combinatorial (multi-vertex) diagrams, providing estimates for when single-vertex results remain reliable (N_max) and detailing collinear-limit corrections via consistency relations, including explicit trispectrum results and associated t_{NL}, tau_{NL}, and g_{NL} scalings. The findings illuminate the structure and detectability of high-point non-Gaussian signatures in axion-monodromy-inspired resonant models, offering concrete predictions for oscillatory N-point shapes in cosmological data.

Abstract

We compute all N-point primordial curvature correlation functions from inflation at tree-level up to N of order ten or more depending on the choice of parameters. This is achieved for resonant inflationary models in which the inflaton potential has a periodic modulation on top of a slow-roll flat term. These models find a natural UV completion in string theory implementation of axion monodromy. Key to the success of our computation is the observation that gravitational interactions among the perturbations can be neglected, which we argue is justified for any model of inflation with parametrically large non-Gaussianity. We provide a comprehensive review and detailed derivations of known consistency relations for squeezed and collinear limits, and generalize them to any N-point function.

Figures (5)

• Figure 1: Single-vertex Feynman diagrams for three- and four-point functions. See Appendix \ref{['Feynman']} for the Feynman rules. At $N=5$, there are three diagrams and so on.
• Figure 2: All tree-level two-vertex diagrams for the trispectrum up to permutations. These diagrams are subleading to the one computed in figure \ref{['contact']} for small $b$ and large $\alpha$. In the collinear limit where $\mathbf{q} \equiv \mathbf{k_1} + \mathbf{k_2} \rightarrow 0$ these diagrams diverge as $q^3$ and will eventually dominate over the single-vertex one in ( \ref{['fullN']}).
• Figure 3: The plot shows $N_{\rm max}$ for different regions of the $\log_{10}(b)-\log_{10}\alpha$ plane. As expected smaller $b$ and larger $\alpha$ extend the validity of ( \ref{['fullN']})-( \ref{['tre']}) to higher and higher $N$'s.
• Figure 4: Propagators and some vertices ($N=3$ and $N=4$) for this theory. Note that $G_A(k,\tau_1,\tau_2) = G_R(k,\tau_2,\tau_1)$. We use the shorthand notation $\delta_{12} = \delta(\tau_1-\tau_2)$. When a two-point function is attached to a vertex, the corresponding time must be integrated over. Internal spatial momenta must also be integrated over $\int d^3p/(2\pi)^3$. As usual, one must also divide by the symmetry factor for the diagram. At $N= 5$, there are 3 diagrams, etc.
• Figure 5: The behavior of an $N$-point function in the limit where an intermediate scalar or graviton propagator has vanishing momenta can be obtained using the consistency relations.