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Scalar field cosmology in phase space

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Abstract

Using dynamical systems methods, we describe the evolution of a minimally coupled scalar field and a Friedmann-Lemaître-Robertson-Walker universe in the context of general relativity, which is relevant for inflation and late-time quintessence eras. Focussing on the spatially flat case, we examine the geometrical structure of the phase space, locate the equilibrium points of the system (de Sitter spaces with a constant scalar field), study their stability through both a third-order perturbation analysis and Lyapunov functions, and discuss the late-time asymptotics. As we do not specify the scalar field’s origin or its potential, the results are independent of the high-energy model.

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Notes

  1. Ref. [46] studies the minimally coupled case of interest here and presents some of the features of scalar field cosmology derived in the following, but its main interest is in initial singularities and particular classes of potentials.

  2. We follow the notations of Ref. [48].

  3. An astronomer would instead choose the density of the matter field \(\Omega _{\phi }\) (in units of the critical density) as another variable.

  4. By contrast, with non-minimally coupled scalar fields, de Sitter spaces with a non-constant scalar field are possible [22, 23].

  5. It is an old adage in cosmological perturbation theory that there are no linear perturbations of de Sitter space sourced by a (minimally coupled) scalar field [70].

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Acknowledgments

We thank Andres Zambrano for discussions and Hugues Beauchesne for help with one of the figures. This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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  1. Physics Department and STAR Research Cluster, Bishop’s University, 2600 College Street, Sherbrooke, QC, J1M 1Z7, Canada

    Valerio Faraoni

  2. Physics Department, Bishop’s University, 2600 College Street, Sherbrooke, QC, J1M 1Z7, Canada

    Charles S. Protheroe

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  1. Valerio Faraoni
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Correspondence to Valerio Faraoni.

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Faraoni, V., Protheroe, C.S. Scalar field cosmology in phase space. Gen Relativ Gravit 45, 103–123 (2013). https://doi.org/10.1007/s10714-012-1462-0

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