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Exact cosmological solutions with nonminimal derivative coupling

Oct, 2009
7 pages
Published in:
  • Phys.Rev.D 80 (2009) 103505
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Abstract: (arXiv)
We consider a gravitational theory of a scalar field ϕ\phi with nonminimal derivative coupling to curvature. The coupling terms have the form κ1Rϕ,μϕ,μ\kappa_1 R\phi_{,\mu}\phi^{,\mu} and κ2Rμνϕ,μϕ,ν\kappa_2 R_{\mu\nu}\phi^{,\mu}\phi^{,\nu} where κ1\kappa_1 and κ2\kappa_2 are coupling parameters with dimensions of length-squared. In general, field equations of the theory contain third derivatives of gμνg_{\mu\nu} and ϕ\phi. However, in the case −2κ1=κ2≡κ-2\kappa_1=\kappa_2\equiv\kappa the derivative coupling term reads κGμνϕ,muϕ,ν\kappa G_{\mu\nu}\phi^{,mu}\phi^{,\nu} and the order of corresponding field equations is reduced up to second one. Assuming −2κ1=κ2-2\kappa_1=\kappa_2, we study the spatially-flat Friedman-Robertson-Walker model with a scale factor a(t)a(t) and find new exact cosmological solutions. It is shown that properties of the model at early stages crucially depends on the sign of κ\kappa. For negative κ\kappa the model has an initial cosmological singularity, i.e. a(t)∼(t−ti)2/3a(t)\sim (t-t_i)^{2/3} in the limit t→tit\to t_i/ and for positive κ\kappa the universe at early stages has the quasi-de Sitter behavior, i.e. a(t)∼eHta(t)\sim e^{Ht} in the limit t→−∞t\to-\infty, where H=(3κ)−1H=(3\sqrt{\kappa})^{-1}. The corresponding scalar field ϕ\phi is exponentially growing at t→−∞t\to-\infty, i.e. ϕ(t)∼e−t/κ\phi(t)\sim e^{-t/\sqrt{\kappa}}. At late stages the universe evolution does not depend on κ\kappa at all/ namely, for any κ\kappa one has a(t)∼t1/3a(t)\sim t^{1/3} at t→∞t\to\infty. Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form κGμνϕ,muϕ,ν\kappa G_{\mu\nu}\phi^{,mu}\phi^{,\nu} is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.
  • 98.80.Cq
  • field theory: scalar
  • coupling: derivative
  • coupling: nonminimal
  • inflation
  • cosmological model
  • gravitation: scalar tensor
  • space-time: Robertson-Walker
  • field equations: solution
  • numerical calculations