Loop Corrections to Cosmological Perturbations in Multi-field Inflationary Models

TL;DR

This paper analyzes one-loop quantum corrections to the adiabatic power spectrum in two-field DBI inflation, focusing on entropy-adiabatic cross-interactions. Using the in-in formalism, it shows that derivative cross-interactions are enhanced by the slow-roll parameter $\ε$ and a small sound speed $c_s$, potentially yielding large loop corrections. Crucially, the calculated loop contributions are infrared finite, owing to the derivative structure of the leading vertices. The results imply that for $c_s \ll 1$ models, higher-order loops and their impact on primordial non-Gaussianities and late-time observables must be carefully accounted for, and a full treatment of the curvature perturbation $\zeta$ via $\delta N$ formalism is needed for connecting to CMB and LSS data.

Abstract

We investigate one-loop quantum corrections to the power spectrum of adiabatic perturbation from entropy modes/adiabatic mode cross-interactions in multiple DBI inflationary models. We find that due to the non-canonical kinetic term in DBI models, the loop corrections are enhanced by slow-varying parameter $ε$ and small sound speed $c_s$. Thus, in general the loop-corrections in multi-DBI models can be large. Moreover, we find that the loop-corrections from adiabatic/entropy cross-interaction vertices are IR finite.

Figures (7)

• Figure 1: Diagrammatic representations of three types of adiabatic-entropy three-point cross-interaction vertices. A red dot denotes temporal derivative while a blue dot denotes spatial derivative, or in fourier space, a momentum factor.
• Figure 2: Diagrammatic representation of the (aa) loop contributions. A "red dot" denotes derivative with respect to comoving time. It is important to note that the left and right end-points are labeled with $\eta_{\ast}$, while the left interaction vertex is labeled with $\eta_1$ and the right vertex is labeled with $\eta_2$.
• Figure 3: Diagrammatic representation of the loop corrections from (bb)-term. As before, a "red dot" denotes derivative with respect to comoving time associated with the vertex, a "blue dot" denotes a momentum factor associated with the line momentum. We may freely label the momentum flows into a vertex a "$+$" sign and momentum flows out of a vertex a "$-$" sign. The momentum factors of the same vertex are "scalar-producted", e.g., the left vertex gives a factor $-\bm{p} \cdot (\bm{p}- \bm{k}_1)$.
• Figure 4: Diagrammatic representation of the loop corrections from (cc)-contribution. As before, a red dot denotes derivative with respect to comoving time associated with the vertex, a blue dot denotes a momentum factor associated with the line momentum. Note that there are two types of diagrams from (cc)-contributions.
• Figure 5: Diagrammatic representations of off-diagonal contributions. From left to right are contributions from (ab), (ac) and (bc) combinations respectively. We do not show the (ba), (ca) and (cb) contributions explicitly.