Integrable Cosmological Potentials

Abstract

The problem of classification of the Einstein--Friedman cosmological Hamiltonians $H$ with a single scalar inflaton field $\phi$ 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 $V(\phi )$. 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.

24 pages, LaTeX, 2 figures; v2: misprints corrected and references added

References in corpus (4)

• Invariant characterization of Liouville metrics and polynomial integrals
• The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta
• General sGoldstino Inflation
• Explicit metrics for a class of two-dimensional superintegrable systems

Cited by in corpus (5)

• Dynamical symmetries in Brans-Dicke cosmology
• Integrable higher-dimensional cosmology with separable variables in an Einstein-dilaton-antisymmetric field theory
• Integrable scalar cosmologies with matter and curvature
• Scalar-tensor cosmologies in a minisuperspace formulation: a case study
• New Integrable Chiral Cosmological Models with Two Scalar Fields