A Hamilton-Jacobi approach to non-slow-roll inflation

TL;DR

The paper develops a non-slow-roll framework for inflation using the Hamilton-Jacobi formalism, treating the equation-of-state parameter $\epsilon(\phi)$ as the central dynamical variable. It derives exact relations connecting $V(\phi)$, $H(\phi)$, and $\epsilon(\phi)$, and shows how slow-roll arises as a limit while providing a useful non-slow-roll solution $\epsilon(\phi) = 3\left(1 - \frac{V(\phi)}{V(\phi_0)}\right)$ for $|\phi - \phi_0| \ll \kappa$. The framework is applied to inverted potentials and to hybrid inflation, yielding explicit $\epsilon(\phi)$, a scalar perturbation amplitude $P_{\cal R}^{1/2}$, and a spectral index $n$ that depends on the branch of the solution. The work enables accurate end-of-inflation calculations and perturbation predictions outside slow-roll, with implications for observational signatures in the CMB.

Abstract

I describe a general approach to characterizing cosmological inflation outside the standard slow-roll approximation, based on the Hamilton-Jacobi formulation of scalar field dynamics. The basic idea is to view the equation of state of the scalar field matter as the fundamental dynamical variable, as opposed to the field value or the expansion rate. I discuss how to formulate the equations of motion for scalar and tensor fluctuations in situations where the assumption of slow roll is not valid. I apply the general results to the simple case of inflation from an ``inverted'' polynomial potential, and to the more complicated case of hybrid inflation.