Exact cosmological solutions with nonminimal derivative coupling

We consider a gravitational theory of a scalar field $\varphi$ 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 $\varphi$. 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. $a(t)\sim (t-t_i)^{2/3}$inthelimit$t\to t_i$;andforpositive$\kappa$ the universe at early stages has the quasi-de Sitter behavior, i.e. $a(t)\sim e^{Ht}$ in the limit $t\to -\mathrm{\infty }$, where $H=(3\sqrt{\kappa }{)}^{-1}$. The corresponding scalar field $\varphi$ is exponentially growing at $t\to -\mathrm{\infty }$, i.e. $\phi(t)\sim e^{-t/\sqrt{\kappa}}$\mathrm{.}Atlatestagestheuniverseevolutiondoesnotdependon$\kappa$atall;namely,forany$\kappa$onehas$a(t)\sim t^{1/3}$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.