On the full quantum trispectrum in multi-field DBI inflation

TL;DR

This work derives the full quantum trispectrum for multi-field DBI inflation by combining contact- and scalar-exchange contributions, and decomposes the result into adiabatic, mixed, and entropic components. It establishes that the squeezed and counter-collinear consistency relations hold, and shows that the trispectrum’s shape depends on the entropy-to-curvature transfer T_RS, enabling discrimination from single-field DBI when transfer is large. The analysis reveals that large T_RS suppresses the tensor-to-scalar ratio r and the equilateral bispectrum f_{NL}^{equil} but enhances the trispectrum amplitude τ_{NL}^{equi} for a fixed f_{NL}^{equil}, yielding distinctive observational signatures. Explicit expressions are provided for both equilateral and non-equilateral configurations, with discussion of how current and future CMB constraints could probe multi-field DBI dynamics.

Abstract

We compute the leading order connected four-point function of the primordial curvature perturbation coming from the four-point function of the fields in multi-field DBI inflation models. We confirm that the consistency relations in the squeezed limit and in the counter-collinear limit hold as in single field models thanks to special properties of the DBI action. We also study the momentum dependence of the trispectra coming from the adiabatic, mixed and purely entropic contributions separately and we find that they have different momentum dependence. This means that if the amount of the transfer from the entropy perturbations to the curvature perturbation is significantly large, the trispectrum can distinguish multi-field DBI inflation models from single field DBI inflation models. A large amount of transfer $T_{\mathcal{RS}} \gg 1 $ suppresses the tensor to scalar ratio $r \propto T_{\mathcal{RS}}^{-2}$ and the amplitude of the bispectrum $f_{NL}^{equi} \propto T_{\mathcal{RS}}^{-2}$ and so it can ease the severe observational constraints on the DBI inflation model based on string theory. On the other hand, it enhances the amplitude of the trispectrum $τ_{NL}^{equi} \propto T_{\mathcal{RS}}^2 f_{NL}^{equi 2}$ for a given amplitude of the bispectrum.

Figures (10)

• Figure 1: The shape of the scalar exchange adiabatic (l.h.s panel) and pure entropy (r.h.s. panel) trispectra as functions of the variables $\cos \theta_1$ and $\cos \theta_2$. The amplitude of the trispectra at the point $\cos\theta_1=\cos\theta_2=-1/3$ has been normalized to unity.
• Figure 2: Plots of the non-linearity parameter $\tau_{NL}^{SE}$ calculated from the scalar exchange adiabatic (l.h.s. panel) and pure entropy (r.h.s.panel) trispectra as functions of the variables $\cos \theta_1$ and $\cos \theta_2$. The amplitude of the first plot was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3$, while the amplitude of the r.h.s. plot was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3/T_{\mathcal{RS}}^4$.
• Figure 3: L.h.s. panel: Plot of the non-linearity parameter $\tau_{NL}^{CI}$ calculated from the contact interaction pure entropy trispectrum as a function of $\cos\theta_1$ and $\cos\theta_2$. R.h.s. panel: Plot of the sum of $\tau_{NL}^{CI}$ and $\tau_{NL}^{SE}$ coming from the pure entropy trispectrum as a function of $\cos\theta_1$ and $\cos\theta_2$. The amplitude of the plots was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3/T_{\mathcal{RS}}^4$.
• Figure 4: Plots of the non-linearity parameter $\tau_{NL}^{SE}$ calculated from the scalar exchange adiabatic (l.h.s. panel) and pure entropy (r.h.s. panel) trispectra for non-equilateral configurations as functions of the momentum amplitude $q$ and $\cos \theta_2$. The amplitude of the first plot was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3$, while the amplitude of the r.h.s. plot was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3/T_{\mathcal{RS}}^4$.
• Figure 5: Plots of the non-linearity parameter $\tau_{NL}^{CI}$ calculated from the contact interaction adiabatic (l.h.s. panel) and pure entropy (r.h.s. panel) trispectra for non-equilateral configurations as functions of the momentum amplitude $q$ and $\cos \theta_2$. The amplitude of the first plot was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3$, while the amplitude of the r.h.s. plot was rescaled by $c_s^4(1+T_{\mathcal{RS}}^2)^3/T_{\mathcal{RS}}^4$.