The geometry of inflationary observables: lifts, flows, equivalence classes

TL;DR

This paper uses the Eisenhart lift to reformulate inflationary dynamics as geodesic motion in a curved field-space, enabling a geometric understanding of inflationary observables independent of specific models.

Contribution

It introduces a geometric formalism for inflation using the Eisenhart lift, expressing observables through properties of a two-dimensional uplifted field-space manifold.

Findings

Inflationary observables can be derived from the geometry of an uplifted field-space.

The formalism abstracts inflation dynamics from specific potentials.

Examples demonstrate the geometric approach's applicability.

Abstract

The Eisenhart lift allows to formulate the dynamics of a scalar field in a potential as pure geodesic motion in a curved field-space manifold involving an additional fictitious vector field. Making use of the formalism in the context of inflation, we show that the main inflationary observables can be expressed in terms of the geometrical properties of a two-dimensional uplifted field-space manifold spanned by the time derivatives of the scalar and the temporal component of the vector. This allows to abstract from specific potentials and models and describe inflation solely in terms of the flow of geometric quantities. Our findings are illustrated through several inflationary examples previously considered in the literature.