A `singular' bounce in the theory of gravity with non-minimal derivative coupling

We explore bounce scenarios in the framework of homogeneous and isotropic cosmological models with arbitrary spatial curvature in the theory of gravity with non-minimal derivative coupling. As expected, we find that there are no turning points and/or bounces in cosmological models with negative or zero spatial curvature. At the same time, both a turning point and a bounce can exist in the model with positive spatial curvature. In particular, the bounce is happened at $\tau=\tau_*$ when $a(\tau_*)=a_{min} =(3\zeta\Omega_2)^{1/2}$, where $\tau=H_0 t$ is a dimensionless cosmic time. It is important fact that the value $a_{min}$ depends {\em only} on $\zeta$ and $\Omega_2$, and does {\em not} depend on $\Omega_0$, $\Omega_3$ and $\Omega_4$. We find that near the bounce $a(\tau)\approx a_{min}(1+\Delta\tau^2/18\zeta)$ and $h(\tau)\approx \Delta\tau/9\zeta$, where $\Delta\tau=\tau-\tau_*$. Thus, the scale factor $a(\tau)$, the Hubble parameter $h(\tau)$, and all corresponding geometrical invariants have a regular behavior near the bounce. As well the values characterizing matter energy densities, such as $\rho_m\sim a^{-3}$ and $\rho_r\sim a^{-4}$, are regular near the bounce. Nevertheless, though the spacetime geometry and energy densities remain to be regular near the bounce, the scalar field has a singular behavior there. Namely, $\phi'\propto 1/\Delta\tau^2 \to\infty$ as $\Delta\tau\to 0$. As a result, we conclude that the complete dynamical system describing the cosmological evolution in theory of gravity with non-minimal derivative coupling is singular near the bounce. On our knowledge, such the scenario, when the spacetime geometry and matter energy densities remain to be regular at approaching the universe evolution to the moment of bounce, while the behavior of scalar field becomes singular, was unknown before. For this reason, we term this scenario as a {\em `singular' bounce}.