Exact cosmological solutions with nonminimal derivative coupling

Abstract

We consider a gravitational theory of a scalar field $\varphi$ with nonminimal derivative coupling to curvature. The coupling terms have the form ${\kappa }_{1}R{\varphi }_{,\mu }{\varphi }^{,\mu }$ and ${\kappa }_2{R}_{\mu \nu }{\varphi }^{,\mu }{\varphi }^{,\nu }$, 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 $\kappa G_{\mu \nu }{\varphi }^{,\mu }{\varphi }^{,\nu }$ 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 depend 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 $\varphi$ is exponentially growing at $t\to -\infty$, i.e., $\varphi (t)\sim e^{-t/\sqrt{\kappa }}$. 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 $\kappa G_{\mu \nu }{\varphi }^{,\mu }{\varphi }^{,\nu }$ is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.