Cosmology with nonminimal kinetic coupling and a Higgs-like potential

Abstract

We consider cosmological dynamics in the theory of gravity with the scalar field possessing the nonminimal kinetic coupling to curvature given as $κG^{μν}ϕ_{,μ}ϕ_{,ν}$, and the Higgs-like potential $V(ϕ)=\fracλ{4}(ϕ^2-ϕ_0^2)^2$. Using the dynamical system method, we analyze stationary points, their stability, and all possible asymptotical regimes of the model under consideration. We show that the Higgs field with the kinetic coupling provides an existence of accelerated regimes of the Universe evolution. There are three possible cosmological scenarios with acceleration: (i) {\em The late-time inflation} when the Hubble parameter tends to the constant value, $H(t)\to H_\infty=(\frac23 πGλϕ_0^4)^{1/2}$ as $t\to\infty$, while the scalar field tends to zero, $ϕ(t)\to 0$, so that the Higgs potential reaches its local maximum $V(0)=\frac14 λϕ_0^4$. (ii) {\em The Big Rip} when $H(t)\sim(t_*-t)^{-1}\to\infty$ and $ϕ(t)\sim(t_*-t)^{-2}\to\infty$ as $t\to t_*$. (iii) {\em The Little Rip} when $H(t)\sim t^{1/2}\to\infty$ and $ϕ(t)\sim t^{1/4}\to\infty$ as $t\to\infty$. Also, we derive modified slow-roll conditions for the Higgs field and demonstrate that they lead to the Little Rip scenario.