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On the full trispectrum in single field DBI-inflation

Frederico Arroja, Shuntaro Mizuno, Kazuya Koyama, Takahiro Tanaka

TL;DR

The paper addresses the trispectrum of primordial curvature perturbations in a general single-field inflation model with non-canonical kinetic terms (P(X,φ)), including DBI and K-inflation. It demonstrates that both contact-interaction and scalar-exchange diagrams contribute at leading order and must be combined to obtain the full trispectrum. For DBI-inflation, in the equilateral configuration, the scalar-exchange contribution dominates, yielding τ_NL^equi ≈ 0.56/c_s^4 with the maximum when the four momenta form equal mutual angles (~109.5°). The work provides explicit CI and SE trispectrum expressions, analyzes their shapes, and discusses observational implications, highlighting the need for dedicated estimators to extract this information from CMB data.

Abstract

We compute the tree-level connected four-point function of the primordial curvature perturbation for a fairly general minimally coupled single field inflationary model, where the inflaton's Lagrangian is a general function of the scalar field and its first derivatives. This model includes K-inflation and DBI-inflation as particular cases. We show that, at the leading order in the slow-roll expansion and in the small sound speed limit, there are two important tree-level diagrams for the trispectrum. One is a diagram where a scalar mode is exchanged and the other is a diagram where the interaction occurs at a point, i.e. a contact interaction diagram. The scalar exchange contribution is comparable to the contact interaction contribution. For the DBI-inflation model, in the so-called equilateral configuration, the scalar exchange trispectrum is maximized when the angles between the four momentum vectors are equal and in this case the amplitude of the trispectrum from the scalar exchange is one order of magnitude higher than the contact interaction trispectrum.

On the full trispectrum in single field DBI-inflation

TL;DR

The paper addresses the trispectrum of primordial curvature perturbations in a general single-field inflation model with non-canonical kinetic terms (P(X,φ)), including DBI and K-inflation. It demonstrates that both contact-interaction and scalar-exchange diagrams contribute at leading order and must be combined to obtain the full trispectrum. For DBI-inflation, in the equilateral configuration, the scalar-exchange contribution dominates, yielding τ_NL^equi ≈ 0.56/c_s^4 with the maximum when the four momenta form equal mutual angles (~109.5°). The work provides explicit CI and SE trispectrum expressions, analyzes their shapes, and discusses observational implications, highlighting the need for dedicated estimators to extract this information from CMB data.

Abstract

We compute the tree-level connected four-point function of the primordial curvature perturbation for a fairly general minimally coupled single field inflationary model, where the inflaton's Lagrangian is a general function of the scalar field and its first derivatives. This model includes K-inflation and DBI-inflation as particular cases. We show that, at the leading order in the slow-roll expansion and in the small sound speed limit, there are two important tree-level diagrams for the trispectrum. One is a diagram where a scalar mode is exchanged and the other is a diagram where the interaction occurs at a point, i.e. a contact interaction diagram. The scalar exchange contribution is comparable to the contact interaction contribution. For the DBI-inflation model, in the so-called equilateral configuration, the scalar exchange trispectrum is maximized when the angles between the four momentum vectors are equal and in this case the amplitude of the trispectrum from the scalar exchange is one order of magnitude higher than the contact interaction trispectrum.

Paper Structure

This paper contains 14 sections, 57 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Non-linear perturbations
  4. Estimate of the amplitude of the total trispectrum
  5. The trispectrum from the contact interaction
  6. The trispectrum from the scalar exchange interaction
  7. $\tau_{NL}$ and the shape of the trispectrum in the equilateral configuration
  8. $\tau_{NL}$: general model
  9. The shape of the trispectrum for the DBI-inflation model in the equilateral configuration
  10. $\tau_{NL}$: DBI-inflation
  11. Conclusion
  12. Canceling the apparent divergences of the trispectrum in the equilateral limit
  13. Where does the maximum of the trispectrum for the DBI-inflation model lie?
  14. The DBI scalar exchange trispectrum in the equilateral limit

Figures (6)

  • Figure 1: Bispectrum diagram
  • Figure 2: On the left: the third order vertex. On the right: the fourth order vertex
  • Figure 3: Trispectrum diagrams. On the left: the contact interaction. On the right: The interaction via exchange of a scalar.
  • Figure 4: Left: The shape of the scalar exchange trispectrum as a function of the variables $\cos \theta_1$ and $\cos \theta_2$. The maximum amplitude has been normalized to unity. Right: Density plot of the shape of the scalar exchange trispectrum as a function of the variables $\cos\theta_1$ and $\cos\theta_2$. The maximum amplitude occurs for $\cos\theta_1=\cos\theta_2=-1/3$.
  • Figure 5: Plot of the non-linearity parameter $\tau_{NL}^{SE}$ calculated from the scalar exchange trispectrum. The maximum amplitude occurs for $\cos\theta_1=\cos\theta_2=-1/3$ and its value was rescaled by $c_s^4$.
  • ...and 1 more figures