Integrable cosmological potentials

The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field φ , which 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 e...

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Bibliographic Details
Published in:Letters in mathematical physics 2017-09, Vol.107 (9), p.1741-1768
Main Authors: Sokolov, V. V., Sorin, A. S.
Format: Article
Language:English
Subjects:
Complex Systems
Economic models
Functions (mathematics)
Geometry
Group Theory and Generalizations
Hamiltonian functions
Integrals
Legendre functions
Mathematical analysis
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
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Summary:The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field φ , which 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 the 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 parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described, and sporadic superintegrable cases are discussed.
ISSN:0377-9017
1573-0530
DOI:10.1007/s11005-017-0962-y