Exact cosmological solutions with nonminimal derivative coupling

We consider a gravitational theory of a scalar field $\ensuremath{\phi}$ with nonminimal derivative coupling to curvature. The coupling terms have the form ${\ensuremath{\kappa}}_{1}R{\ensuremath{\phi}}_{,\ensuremath{\mu}}{\ensuremath{\phi}}^{,\ensuremath{\mu}}$ and ${\ensuremath{\kappa}}_{2}{R}_{\ensuremath{\mu}\ensuremath{\nu}}{\ensuremath{\phi}}^{,\ensuremath{\mu}}{\ensuremath{\phi}}^{,\ensuremath{\nu}}$, where ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ are coupling parameters with dimensions of length squared. In general, field equations of the theory contain third derivatives of ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ and $\ensuremath{\phi}$. However, in the case $\ensuremath{-}2{\ensuremath{\kappa}}_{1}={\ensuremath{\kappa}}_{2}\ensuremath{\equiv}\ensuremath{\kappa}$, the derivative coupling term reads $\ensuremath{\kappa}{G}_{\ensuremath{\mu}\ensuremath{\nu}}{\ensuremath{\phi}}^{,\ensuremath{\mu}}{\ensuremath{\phi}}^{,\ensuremath{\nu}}$ and the order of corresponding field equations is reduced up to second one. Assuming $\ensuremath{-}2{\ensuremath{\kappa}}_{1}={\ensuremath{\kappa}}_{2}$, we study the spatially-flat Friedman-Robertson-Walker model with a scale factor $a(t)$ and find new exact cosmological solutions. It is shown that properties of the model at early stages crucially depend on the sign of $\ensuremath{\kappa}$. For negative $\ensuremath{\kappa}$, the model has an initial cosmological singularity, i.e., $a(t)\ensuremath{\sim}(t\ensuremath{-}{t}_{i}{)}^{2/3}$ in the limit $t\ensuremath{\rightarrow}{t}_{i}$; and for positive $\ensuremath{\kappa}$, the Universe at early stages has the quasi-de Sitter behavior, i.e., $a(t)\ensuremath{\sim}{e}^{Ht}$ in the limit $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, where $H=(3\sqrt{\ensuremath{\kappa}}{)}^{\ensuremath{-}1}$. The corresponding scalar field $\ensuremath{\phi}$ is exponentially growing at $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, i.e., $\ensuremath{\phi}(t)\ensuremath{\sim}{e}^{\ensuremath{-}t/\sqrt{\ensuremath{\kappa}}}$. At late stages, the Universe evolution does not depend on $\ensuremath{\kappa}$ at all; namely, for any $\ensuremath{\kappa}$ one has $a(t)\ensuremath{\sim}{t}^{1/3}$ at $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form $\ensuremath{\kappa}{G}_{\ensuremath{\mu}\ensuremath{\nu}}{\ensuremath{\phi}}^{,\ensuremath{\mu}}{\ensuremath{\phi}}^{,\ensuremath{\nu}}$ is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.