Integrable Cosmological Potentials

TL;DR

This paper classifies Einstein-Friedman cosmological models with a scalar inflaton field that admit polynomial integrals of motion, deriving conditions for their existence and explicitly constructing several classes of integrable potentials.

Contribution

It provides necessary and sufficient conditions for polynomial integrals in cosmological Hamiltonians and constructs new integrable potentials, including a wide class with cubic integrals.

Findings

Derived conditions for existence of polynomial integrals in cosmological Hamiltonians.

Explicit solutions for linear and quadratic integrals.

Identified a new class of Hamiltonians with cubic integrals and their potentials.

Abstract

The problem of classification of the Einstein--Friedman cosmological Hamiltonians $H$ with a single scalar inflaton field $\varphi$ that possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint $H=0$ is considered. Necessary and sufficient conditions for the existence of first, second, and third degree integrals are derived. These conditions have the form of ODEs for the cosmological potential $V(\varphi)$. In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in a parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described and sporadic superintegrable cases are discussed.