Cosmological dynamics of non-minimally coupled scalar field system and its late time cosmic relevance

We investigate the cosmological dynamics of non-minimally coupled scalar field system described by $F(\phi)R$ coupling with $F(\phi)=(1-\xi\phi^N)R$($N\ge2$) and the field potential, $V(\phi)=V_0\phi^n$. We use a generic set of dynamical variables to bring out new asymptotic regimes of the underlying dynamics. However, our dynamical variables miss the most important fixed point$-$ the de Sitter solution. We make use of the original form of system of equations to investigate the issues related to this important solution. In particular, we show that the de-Sitter solution which is a dynamical attractor of the system lies in the region of negative effective gravitational constant $G_N$ thereby leading to a ghost dominated universe in future and a transient quintessence(phantom) phase with $G_N>0 $ around the present epoch (however, as demonstrated by Starobinsky in 1981, the ghost dominated universe, if exists, can not be accessed from the Universe we live in, we shall say more about this important result in the last section). We also carry out comparison of the model with other competing models of dark energy such as galileon modified gravity and others.