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Cosmology with nonminimal kinetic coupling and a power-law potential

We consider cosmological dynamics in the theory of gravity with the scalar field possessing a nonminimal kinetic coupling to gravity, $κG_{μν}ϕ^μϕ^ν$, and the power-law potential $V(ϕ)=V_0ϕ^N$. Using the dynamical system method, we analyze all possible asymptotical regimes of the model under investigation and show that for sloping potentials with $0<N<2$ there exists a quasi-de Sitter asymptotic $H={1}/{\sqrt{9κ}}$ corresponding to an early inflationary Universe. In contrast to the standard inflationary scenario, the kinetic coupling inflation does not depend on a scalar field potential and is only determined by the coupling parameter $κ$. We obtain that there exist two different late-time asymptotical regimes. The first one leads to the usual power-like cosmological evolution with $H=1/3t$, while the second one represents the late-time inflationary Universe with $H=1/\sqrt{3κ}$. This secondary inflationary phase depends only on $κ$ and is a specific feature of the model with nonminimal kinetic coupling. Additionally, an asymptotical analysis shows that for the quadric potential with N=2 the asymptotical regimes remain qualitatively the same, while the kinetic coupling inflation is impossible for steep potentials with N>2. Using a numerical analysis, we also construct exact cosmological solutions and find initial conditions leading to the initial kinetic coupling inflation followed either by a &#34;graceful&#34; oscillatory exit or by the secondary inflation.