A geometric description of the non-Gaussianity generated at the end of multi-field inflation

TL;DR

This paper presents a geometric framework for understanding the non-Gaussianity generated at the end of multi-field inflation, linking the non-Gaussianity parameters to the curvature properties of the end hypersurface.

Contribution

It introduces a geometric description of non-Gaussianity in multi-field inflation, relating parameters to the curvature of the end-of-inflation hypersurface.

Findings

Non-Gaussianity parameters are determined by the geometry of the end hypersurface.

f_NL is proportional to the curvature of the curve on the hypersurface.

g_NL relates to the change in curvature radius along the curve.

Abstract

In this paper we mainly focus on the curvature perturbation generated at the end of multi-field inflation, such as the multi-brid inflation. Since the curvature perturbation is produced on the super-horizon scale, the bispectrum and trispectrum have a local shape. The size of bispectrum is measured by $f_{NL}$ and the trispectrum is characterized by two parameters $\tau_{NL}$ and $g_{NL}$. For simplicity, the trajectory of inflaton is assumed to be a straight line in the field space and then the entropic perturbations do not contribute to the curvature perturbation during inflation. As long as the background inflaton path is not orthogonal to the hyper-surface for inflation to end, the entropic perturbation can make a contribution to the curvature perturbation at the end of inflation and a large local-type non-Gaussiantiy is expected. An interesting thing is that the non-Gaussianity parameters are completely determined by the geometric properties of the hyper-surface of the end of inflation. For example, $f_{NL}$ is proportional to the curvature of the curve on this hyper-surface along the adiabatic direction and $g_{NL}$ is related to the change of the curvature radius per unit arc-length of this curve. Both $f_{NL}$ and $g_{NL}$ can be positive or negative respectively, but $\tau_{NL}$ must be positive and not less than $({6\over 5}f_{NL})^2$.