Large Primordial Trispectra in General Single Field Inflation

TL;DR

This paper classifies the leading primordial trispectra in general single-field inflation with a P(X, φ) Lagrangian, showing four equilateral-shaped configurations controlled by the parameters λ/Σ, μ/Σ, and 1/c_s^2, and provides simple templates for data analyses. It computes the trispectrum from both scalar-exchange and contact-interaction diagrams, yielding six shape functions that combine into a total T with six contributing terms, and discusses how these shapes differ from local-trispectrum forms. The authors also explore the impact of non-Bunch-Davies initial states, finding significant enhancements and folded-limit divergences that require a cutoff, highlighting trispectra as a sensitive probe of initial-state physics. The work lays a foundation for using trispectrum templates to constrain inflationary interactions and potential new physics with upcoming CMB, large-scale structure, and 21-cm surveys.

Abstract

We compute the large scalar four-point correlation functions in general single field inflation models, where the inflaton Lagrangian is an arbitrary function of the inflaton and its first derivative. We find that the leading order trispectra have four different shapes determined by three parameters. We study features in these shapes that can be used to distinguish among themselves, and between them and the trispectra of the local form. For the purpose of data analyses, we give two simple representative forms for these "equilateral trispectra". We also study the effects on the trispectra if the initial state of inflation deviates from the standard Bunch-Davies vacuum.

Figures (9)

• Figure 1: Two diagrams that contribute to the large trispectra.
• Figure 2: This figure illustrates the tetrahedron we consider.
• Figure 3: In this group of figures, we consider the equilateral limit $k_1=k_2=k_3=k_4$, and plot $T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$, $T_{loc1}$ and $T_{loc2}$, respectively, as functions of $k_{12}/k_1$ and $k_{14}/k_1$. Note that $T_{loc1}$ blows up when $k_{12}\ll k_1$ and $k_{14} \ll k_1$. $T_{loc1}$ also blows up in the other boundary, because this boundary corresponds to $k_{13}\ll k_1$. So $T_{loc1}$ is distinguishable from all other shapes in this limit. We also note that $T_{c1}$ and $T_{loc2}$ are both independent of $k_{12}$ and $k_{14}$.
• Figure 4: In this group of figures, we consider the folded limit $k_{12}=0$, and plot $T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$ and $T_{loc2}$, respectively, as functions of $k_{14}/k_1$ and $k_{4}/k_1$. $T_{loc1}$ blows up in this limit. Note that when $k_4\rightarrow 0$, all shape functions except $T_{loc1}$ and $T_{loc2}$ vanish.
• Figure 5: In this group of figures, we consider the specialized planar limit with $k_1=k_3=k_{14}$, and plot $T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$, $T_{loc1}$ and $T_{loc2}$, respectively, as functions of $k_{2}/k_1$ and $k_{4}/k_1$. Again, in the $k_2\rightarrow 0$ or $k_4\rightarrow 0$ limit, our shape functions vanish as ${\cal O}(k_2^2)$ and ${\cal O}(k_4^2)$ respectively. This is different from that of the local shape. $T_{loc1}$ blows up when $k_2 \rightarrow k_4$. This is because in this limit, $k_{13}\rightarrow 0$.