Primordial Non-Gaussianity in Multi-Scalar Inflation

TL;DR

This work derives a concise, δN-formalism-based formula for the primordial non-Gaussianity parameter $f_{NL}$ in multi-field inflation without assuming slow-roll. It expresses $f_{NL}$ in terms of first- and second-order derivatives of the number of e-folds and a trajectory-dependent object $\Theta^a$, enabling computation from linear perturbations even when slow-roll is temporarily violated. The paper applies the framework to a two-field double inflation with a large mass ratio and shows that, although $f_{NL}$ can transiently reach ${\cal O}(1)$, its final value is suppressed by slow-roll parameters evaluated at horizon exit, typically yielding $f_{NL}\ll 1$; analytical expressions in the large-mass limit corroborate the numerical results. Extending to ${\cal N}$-flation, the analysis suggests a general suppression of non-Gaussianity, scaling roughly as $f_{NL}\sim 1/(2N)$, aligning with slow-roll expectations and indicating limited scope for large primordial non-Gaussianity in these multi-field scenarios. The framework also accommodates general field-space metrics, reducing computational complexity from ${\cal O}(\mathcal{N}^2)$ to ${\cal O}(\mathcal{N})$ and providing a practical tool for exploring non-Gaussian signatures in models with many fields.

Abstract

We give a concise formula for the non-Gaussianity of the primordial curvature perturbation generated on super-horizon scales in multi-scalar inflation model without assuming slow-roll conditions. This is an extension of our previous work. Using this formula, we study the generation of non-Gaussianity for the double inflation models in which the slow-roll conditions are temporarily violated after horizon exit, and we show that the non-linear parameter $f_{NL}$ for such models is suppressed by the slow-roll parameters evaluated at the time of horizon exit.

Figures (6)

• Figure 1: This figure shows a background trajectory in field space. We set initial values of fields to $\phi_*=\chi_*=10$.
• Figure 2: (color online) This figure shows evolution of $\phi$ (dashed red line) and $\chi$ (solid blue line). We choose the $e$-folding number, $N$, as a time coordinate and set the initial time, $N_*$, to $0$.
• Figure 3: This figure shows the evolution of the slow-roll parameter $\epsilon$ defined by Eq. (\ref{['epsilon']}).
• Figure 4: This figure shows the evolution of the slow-roll parameter $\eta_{\phi\phi}$ defined by Eq. (\ref{['eta']}).
• Figure 5: This figure shows the evolution of the slow-roll parameter $\eta_{\chi\chi}$ defined by Eq. (\ref{['eta']}).