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

Qing-Guo Huang

TL;DR

The paper addresses how local-type non-Gaussianity can arise at the end of multi-field inflation and demonstrates that the amplitudes of $f_{NL}$, $\tau_{NL}$, and $g_{NL}$ are determined by the geometry of the end-of-inflation hypersurface. By applying the $\delta N$ formalism to two-field and then $n$-field slow-roll models with a straight inflaton path, it derives explicit relations between the non-Gaussianity parameters and geometric quantities such as the extrinsic curvature $\kappa$ of the end curve (or surface) and its variations, including contributions from transverse entropic directions. The results show that $f_{NL}$ and $g_{NL}$ vanish when the end-curve is straight ($\kappa=0$), while $\tau_{NL}$ remains generally positive and obeys $\tau_{NL} \ge (6/5 f_{NL})^2$, with richer behavior in multi-field cases due to additional entropic directions. This work provides a geometric bridge between observable non-Gaussianity and the end-of-inflation geometry, offering a pathway to potentially infer aspects of the high-dimensional field-space structure relevant to string-inspired inflation scenarios.

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 $τ_{NL}$ 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 $τ_{NL}$ must be positive and not less than $({6\over 5}f_{NL})^2$.

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

TL;DR

The paper addresses how local-type non-Gaussianity can arise at the end of multi-field inflation and demonstrates that the amplitudes of , , and are determined by the geometry of the end-of-inflation hypersurface. By applying the formalism to two-field and then -field slow-roll models with a straight inflaton path, it derives explicit relations between the non-Gaussianity parameters and geometric quantities such as the extrinsic curvature of the end curve (or surface) and its variations, including contributions from transverse entropic directions. The results show that and vanish when the end-curve is straight (), while remains generally positive and obeys , with richer behavior in multi-field cases due to additional entropic directions. This work provides a geometric bridge between observable non-Gaussianity and the end-of-inflation geometry, offering a pathway to potentially infer aspects of the high-dimensional field-space structure relevant to string-inspired inflation scenarios.

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 and the trispectrum is characterized by two parameters and . 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, is proportional to the curvature of the curve on this hyper-surface along the adiabatic direction and is related to the change of the curvature radius per unit arc-length of this curve. Both and can be positive or negative respectively, but must be positive and not less than .

Paper Structure

This paper contains 4 sections, 118 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Two-field slow-roll inflation
  3. $n$-field slow-roll inflation
  4. Discussions

Figures (4)

  • Figure 1: The green straight line is the trajectory of inflaton during inflation and the black curve corresponds the field configuration at the end of inflation.
  • Figure 2: The green straight line is the trajectory of inflaton during inflation and the curve corresponds the field configuration at the end of inflation.
  • Figure 3: The green straight line is the trajectory of inflaton fields during inflation and the curve $C$ corresponds the field configuration at the end of inflation. $C$ is a straight line in (b).
  • Figure 4: The green line is the trajectory of inflaton fields during inflation and the hyper-surface corresponds the field configuration at the end of inflation.