Testing Multi-Field Inflation: A Geometric Approach

TL;DR

This paper introduces a geometric framework for analyzing multifield inflation models, linking spectral features to the underlying field space geometry and kinematics, and providing tools to distinguish models with different numbers of active fields.

Contribution

It develops a geometric approach connecting inflationary spectra to field space geometry, and introduces observables to differentiate between two-field and multi-field inflation scenarios.

Findings

Mode conservation laws analogous to single-field inflation.

Mode sourcing relations depend on potential derivatives and field metric.

A multifield observable can distinguish between different numbers of active fields.

Abstract

We develop an approach for linking the power spectra, bispectrum, and trispectrum to the geometric and kinematical features of multifield inflationary Lagrangians. Our geometric approach can also be useful in determining when a complicated multifield model can be well approximated by a model with one, two, or a handful of fields. To arrive at these results, we focus on the mode interactions in the kinematical basis, starting with the case of no sourcing and showing that there is a series of mode conservation laws analogous to the conservation law for the adiabatic mode in single-field inflation. We then treat the special case of a quadratic potential with canonical kinetic terms, showing that it produces a series of mode sourcing relations identical in form to that for the adiabatic mode. We build on this result to show that the mode sourcing relations for general multifield inflation are extension of this special case but contain higher-order covariant derivatives of the potential and corrections from the field metric. In parallel, we show how these interactions depend on the geometry of the inflationary Lagrangian and on the kinematics of the associated field trajectory. Finally, we consider how the mode interactions and effective number of fields active during inflation are reflected in the spectra and introduce a multifield consistency relation, as well as a multifield observable that can potentially distinguish two-field scenarios from scenarios involving three or more effective fields.