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Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology

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Abstract

We discuss the Hamiltonian dynamics for cosmologies coming from Extended Theories of Gravity. In particular, minisuperspace models are taken into account searching for Noether symmetries. The existence of conserved quantities gives selection rule to recover classical behavior in cosmic evolution according to the so-called Hartle criterion, which allows one to select correlated regions in the configuration space of dynamical variables. We show that such a statement works for general classes of Extended Theories of Gravity and is conformally preserved. Furthermore, the presence of Noether symmetries allows a straightforward classification of singularities that represent the points where the symmetry is broken. Examples for non-minimally coupled and higher-order models are discussed.

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Notes

  1. Up to now, there is no final evidence that dark energy and dark matter exist at fundamental quantum level [47].

  2. We adopt the conventions: μ,ν=0,1,2,3 for the Greek space-time indices and i,j=1,2,3 for the purely spatial indices.

  3. In this case, the Hamiltonian is a constraint being a sum of constraints where the lapse and shift functions play the role of Lagrange multipliers.

  4. We indicate the quantities as Lagrangians and vector fields with a tilde if the non-degenerate transformation

    $$Q^i=Q^i(q) ,\qquad \dot{Q}^i(q)= \frac{\partial Q^i}{\partial q^j}\dot{q}^j, $$

    is performed. However, the Jacobian determinant \(\mathcal{J}=\Vert\partial Q^{i} /\partial q^{j}\Vert\) has to be non-zero.

  5. See [89–91] for the possibility of continuation beyond singular points of the conformal factor.

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Acknowledgements

S.C. wants to acknowledge G. Lambiase and C. Stornaiolo for useful discussions and comments on topics related with this paper. M.D.L. has been partially supported by INFN-MEC bilateral agreement and iniziativa specifica NA12-INFN. The work by S.D.O. has been supported in part by MICINN (Spain) project FIS2010-15640, by AGAUR 2009SGR-994 project.

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Authors and Affiliations

  1. Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”, Edificio G, Via Cinthia, 80126, Napoli, Italy

    S. Capozziello & M. De Laurentis

  2. INFN sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, 80126, Napoli, Italy

    S. Capozziello & M. De Laurentis

  3. Istitucio Catalana de Recerca i Estudis Avancats (ICREA) and Consejo Superior de Investigaciones Cientificas, ICE/CSIC and Institut de Ciencies de l Espai (IEEC-CSIC), Campus UAB, Facultat de Ciencies, Torre C5-Par-2a pl, 08193, Bellaterra, Barcelona, Spain

    S. D. Odintsov

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  1. S. Capozziello
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Capozziello, S., De Laurentis, M. & Odintsov, S.D. Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology. Eur. Phys. J. C 72, 2068 (2012). https://doi.org/10.1140/epjc/s10052-012-2068-0

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