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arXiv · 1608.08511

Integrable Cosmological Potentials

V. V. Sokolov·A. S. Sorin
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Abstract

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

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V. V. Sokolov, A. S. Sorin. 2016-09-14. Integrable Cosmological Potentials. https://doi.org/10.1007/s11005-017-0962-y

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