Primordial Non-Gaussianities from the Trispectra in Multiple Field Inflationary Models

TL;DR

This work derives the leading trispectrum of the curvature perturbation in general multi-field inflation with a Lagrangian $P(X^{IJ},\phi^I)$, capturing both adiabatic and entropy perturbations in the small sound-speed regime. It identifies three fundamental momentum-space shape functions $A_1$, $A_2$, and $A_3$ arising from distinct four-point contact interactions, and shows the observable trispectrum $T_{\mathcal{R}}$ is fully determined by the speeds $c_a$, $c_e$, the nonlinear parameters $\lambda$, $\Pi$, and the superhorizon transfer $T_{\mathcal{R}\mathcal{S}}$. By decomposing perturbations into adiabatic and entropy modes and propagating them through horizon crossing to late times, the paper provides explicit expressions for the quartic interactions and the resulting trispectra in the inflaton fields and in $\mathcal{R}$. It further develops a framework to visualize and quantify the trispectrum via shape diagrams and a generalized $\tau_{\mathrm{NL}}$ parameter, highlighting how multi-field dynamics introduce richer, degenerate shape information than single-field models. The results generalize previous single-field and multi-field DBI findings and set the stage for confronting trispectrum shapes with observations, while noting assumptions such as vanishing cross-correlations at horizon crossing and the focus on contact interactions.

Abstract

We investigate the primordial non-Gaussianities from the trispectra in multi-field inflation models, which can be seen as generalization of multi-field $k$-inflation and multi-DBI inflation. We derive the full fourth-order perturbation action for the inflaton fields and evaluate the four-point correlation functions for the perturbations in the limit $\ca \ll 1$ and $\ce \ll1$. There are three types of momentum-dependent shape functions which arise from three types of four-point interaction vertices. The final trispectrum of the curvature perturbation can be expressed in terms of the deformations and permutations of these three shape functions, and is determined by $\ca$, $\ce$, $λ$, $Π$ which depend on the non-linear structure of the model and also the transfer function $T_{\Rc\Sc}$. We also discuss the parameter space for the trispectrum and plot the shape diagrams for the trispectrum both for visualization and for distinguishing different shapes from each other.

Figures (6)

• Figure 1: Diagrammatic representations of the four-point adiabatic mode self-interaction vertices. A "red dot" denotes derivative with respect to comoving time $\eta$. A blue dot denotes derivative with respect to spatial coordinates, or in Fourier space the spatial momentum. A line between two blue dots denotes "scalar-product".
• Figure 2: Diagrammatic representations of the four-point adiabatic/entropy modes cross-interaction vertices. A "red dot" denotes derivative wrt comoving time $\eta$. A blue dot donotes derivative wrt spatial coordinates, or in Fourier space the spatial momentum. A line between two blue dots denotes "scalar-product".
• Figure 3: Diagrammatic representations of the four-point entropy mode self-interaction vertices. A "red dot" denotes derivative wrt comoving time $\eta$. A blut dot donotes derivative wrt spatial coordinates, or in fourier space the spatial momentum. A line between two blue dots denotes "scalar-product".
• Figure 6: Shapes of $\mathcal{A}_1$, $\mathcal{A}_2$ and $\mathcal{A}_3$ with momentum configuration $k_1=k_2=k_{12}=k_{23}$ as function of $\theta_1$ and $\theta_2$. In this group of diagrams, parameters are chosen as $c_{\textrm{a}}=c_{\textrm{e}}=0.1$, $\epsilon=0.01$, $\lambda=\Pi=100$ and $T_{\mathcal{R}\mathcal{S}}=1$.
• Figure 7: Shapes of $\mathcal{A}_1$, $\mathcal{A}_2$ and $\mathcal{A}_3$ with 2D momentum configuration $k_1=k_2=k_{12}$ as function of $k_1$ and $k_2$. In this group of diagrams, parameters are chosen as $c_{\textrm{a}}=c_{\textrm{e}}=0.1$, $\epsilon=0.01$, $\lambda=\Pi=100$ and $T_{\mathcal{R}\mathcal{S}}=1$.