Non-Gaussianity in Two-Field Inflation

TL;DR

This paper derives semi-analytic formulas for non-Gaussianity in two-field inflation, linking observable non-Gaussian features to the inflationary potential's geometry and initial conditions, and explains why large non-Gaussianity is rare.

Contribution

It provides a geometric and semi-analytic framework to connect non-Gaussianity in two-field inflation with initial conditions and potential features, highlighting conditions for observable non-Gaussianity.

Findings

Large non-Gaussianity requires sensitive sourcing of curvature by isocurvature modes.

Trajectories diverging in field space lead to potential for large non-Gaussianity.

Attractor solutions in two-field inflation generally produce small non-Gaussianity.

Abstract

We derive semi-analytic formulae for the local bispectrum and trispectrum in general two-field inflation and provide a simple geometric recipe for building observationally allowed models with observable non-Gaussianity. We use the \delta N formalism and the transfer function formalism to express the bispectrum almost entirely in terms of model-independent physical quantities. Similarly, we calculate the trispectrum and show that the trispectrum parameter \tau NL can be expressed entirely in terms of spectral observables, which provides a new consistency relation unique to two-field inflation. We show that in order to generate observably large non-Gaussianity during inflation, the sourcing of curvature modes by isocurvature modes must be extremely sensitive to the initial conditions, and that the amount of sourcing must be moderate in order to avoid excessive fine-tuning. Under some minimal assumptions, we argue that the first condition is satisfied only when neighboring trajectories through the two-dimensional field space diverge during inflation. Geometrically, this means that the inflaton must roll along a ridge in the potential V for some time during inflation and that its trajectory must turn slightly (but not too sharply) in field space. Therefore, it follows that two-field scenarios with attractor solutions necessarily produce small non-Gaussianity. This explains why it has been so difficult to achieve large non-Gaussianity in two-field inflation, and why it has only been achieved in a narrow class of models like hybrid inflation and certain product potentials where the potential and/or the initial conditions are fine-tuned. Some of our conclusions generalize qualitatively to general multi-field inflation.