The superpotential method in cosmological inflation

TL;DR

The paper introduces the superpotential method as an efficient alternative to traditional equations in cosmological inflation, enabling direct calculation of observables and systematic slow-roll expansion analysis.

Contribution

It demonstrates that the superpotential equation simplifies inflationary calculations and provides a new approach for analyzing inflationary models.

Findings

Allows direct computation of inflationary observables from the superpotential

Enables systematic and arbitrary order slow-roll expansion

Facilitates estimation of the convergence radius of the slow-roll series

Abstract

Scalar field cosmological inflation has a first integral relating the Hubble function and the lagrangian of the scalar field(s), which is known under the names of "Hamilton-Jacobi approach" or "superpotential equation". Here we exploit the simplicity of this superpotential equation and use it as an alternative but equivalent cosmological evolution equation during inflation, replacing the Friedman-Robertson-Walker (FRW) equations. It turns out that all inflationary observables can be calculated directly from its solution (the superpotential). Further, the superpotential equation allows for a simple and direct calculation of the slow-roll expansion to arbitrary order and, in many cases, for an exact determination of the slow-roll attractor. It also allows for a power series expansion in the inflaton field which permits to estimate the radius of convergence of the slow-roll expansion. We consider several examples of single-field inflationary models to demonstrate the simplicity and usefulness of the method.