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  • 1.
    Alho, Artur
    et al.
    Univ Lisbon, Ctr Math Anal Geometry & Dynam Syst, Inst Super Tecn, Av Rovisco Pais, P-1049001 Lisbon, Portugal..
    Carloni, Sante
    Univ Lisbon, Ctr Multidisciplinar Astrofis CENTRA, Inst Super Tecn, Av Rovisco Pais, P-1049001 Lisbon, Portugal..
    Uggla, Claes
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).
    On dynamical systems approaches and methods in f (R) cosmology2016In: Journal of Cosmology and Astroparticle Physics, ISSN 1475-7516, E-ISSN 1475-7516, no 8, article id 064Article in journal (Refereed)
    Abstract [en]

    We discuss dynamical systems approaches and methods applied to flat Robertson Walker models in f(R)-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, f(R) = R + alpha R-2, alpha > 0, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to f (R) cosmology and discuss their advantages and disadvantages.

  • 2.
    Alho, Artur
    et al.
    Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Portugal.
    Uggla, Claes
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).
    Inflationary alpha-attractor cosmology: A global dynamical systems perspective2017In: Physical Review D: covering particles, fields, gravitation, and cosmology, ISSN 2470-0010, E-ISSN 2470-0029, Vol. 95, no 8, article id 083517Article in journal (Refereed)
    Abstract [en]

    We study flat Friedmann-Lemaitre-Robertson-Walker alpha-attractor E- and T-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of e-folds, is associated with a particular solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this "inflationary attractor solution." A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.

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