Exact cosmological solutions with nonminimal derivative coupling

We consider a gravitational theory of a scalar field $\phi$ with nonminimal derivative coupling to curvature. The coupling terms have the form $\kappa_1 R\phi_{,\mu}\phi^{,\mu}$ and $\kappa_2 R_{\mu\nu}\phi^{,\mu}\phi^{,\nu}$ where $\kappa_1$ and $\kappa_2$ are coupling parameters with dimensions of length-squared. In general, field equations of the theory contain third derivatives of $g_{\mu\nu}$ and $\phi$. However, in the case $-2\kappa_1=\kappa_2\equiv\kappa$ the derivative coupling term reads $\kappa G_{\mu\nu}\phi^{,mu}\phi^{,\nu}$ and the order of corresponding field equations is reduced up to second one. Assuming $-2\kappa_1=\kappa_2$, 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 depends on the sign of $\kappa$. For negative $\kappa$ the model has an initial cosmological singularity, i.e. $a(t)\sim (t-t_i)^{2/3}$ in the limit $t\to t_i$; and for positive $\kappa$ the universe at early stages has the quasi-de Sitter behavior, i.e. $a(t)\sim e^{Ht}$ in the limit $t\to-\infty$, where $H=(3\sqrt{\kappa})^{-1}$. The corresponding scalar field $\phi$ is exponentially growing at $t\to-\infty$, i.e. $\phi(t)\sim e^{-t/\sqrt{\kappa}}$. At late stages the universe evolution does not depend on $\kappa$ at all; namely, for any $\kappa$ one has $a(t)\sim t^{1/3}$ at $t\to\infty$. Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form $\kappa G_{\mu\nu}\phi^{,mu}\phi^{,\nu}$ is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.