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Exact cosmological solutions with nonminimal derivative coupling

We consider a gravitational theory of a scalar field $ϕ$ with nonminimal derivative coupling to curvature. The coupling terms have the form $κ_1 Rϕ_{,μ}ϕ^{,μ}$ and $κ_2 R_{μν}ϕ^{,μ}ϕ^{,ν}$ where $κ_1$ and $κ_2$ are coupling parameters with dimensions of length-squared. In general, field equations of the theory contain third derivatives of $g_{μν}$ and $ϕ$. However, in the case $-2κ_1=κ_2\equivκ$ the derivative coupling term reads $κG_{μν}ϕ^{,mu}ϕ^{,ν}$ and the order of corresponding field equations is reduced up to second one. Assuming $-2κ_1=κ_2$, 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 $κ$. For negative $κ$ 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 $κ$ 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 $H=(3\sqrtκ)^{-1}$. The corresponding scalar field $ϕ$ is exponentially growing at $t\to-\infty$, i.e. $ϕ(t)\sim e^{-t/\sqrtκ}$. At late stages the universe evolution does not depend on $κ$ at all; namely, for any $κ$ 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 $κG_{μν}ϕ^{,mu}ϕ^{,ν}$ is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.